# 统计代写|主成分分析代写Principal Component Analysis代考|Title: Fundamental Principles of Finite Element Method

The passage describes the fundamental principles of the finite element method (FEM) as applied to solving engineering problems involving continuous solid bodies, often with complex geometries and behaviors governed by partial differential equations. Line elements like springs, trusses, and beams serve as simple examples to introduce the core processes involved in formulating and solving FEM problems.

The accuracy of a finite element solution is assessed by studying its convergence as the computational mesh is refined. There are two main approaches to refining the mesh:

h-refinement: By subdividing existing elements or adding more elements to cover the domain, thereby reducing the size of individual elements. This process aims to capture finer variations in the field variables by increasing the resolution of the mesh.

p-refinement: Without changing the element sizes, the polynomial degree of the interpolation functions is increased. This enhances the approximation capability of each element without necessarily increasing their number.

Solution convergence is demonstrated when a sequence of increasingly refined solutions converges asymptotically towards the exact solution. Although mathematical proofs of convergence typically assume regular mesh refinement, real-world applications often employ unstructured meshes due to the irregularity of the modeled geometries. Modern finite element software tools are adept at generating such meshes automatically.

The example provided illustrates the convergence of the solution for a rectangular elastic plate loaded at a corner. It uses h-refinement with rectangular plane stress elements, showing that as the mesh is refined, the computed maximum normal stress converges to a value close to the exact solution derived from elementary beam theory.

The interpolation functions used to represent the field variable across elements must adhere to two key conditions:

Compatibility Requirement: To ensure solution convergence, along element boundaries, the field variable and its partial derivatives up to one order less than the highest-order derivative in the integral formulation of the element equations must be continuous. For instance, if the first derivative of displacement appears in the element equation, as in the truss element, displacement continuity is required, but not necessarily its derivatives. However, for the beam element, where the second derivative of displacement appears, both displacement and its first derivative (slope) must be continuous.

Completeness Requirement: In the limit of infinitely small elements, the field variable and its partial derivatives up to and including the highest-order derivative must be able to assume constant values throughout the element. This criterion allows for the representation of rigid body motions (displacement), rotations (beam slope), and uniform states (temperature, heat flux, strain, etc.), thus ensuring the finite element can accommodate the full range of possible physical behaviors.

The passage acknowledges that the discussion is not exhaustive and points readers to referenced texts for a deeper dive into the theoretical underpinnings of convergence and interpolation function requirements in finite element analysis. These requirements guide the development of suitable interpolation functions for various element shapes and complexities encountered in practical engineering applications.

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